1^, 



j = 1,2, 



(2-82) 



may be thought of as defining the wavelengths capable of causing resonance in 

 a basin of length £g. The general form of equation (2-81) is found by 

 substituting equation (2-82) into the expression for the wavelength; there- 

 fore. 



T = 



kTil 



-1 1/2 



B 



jg tanh(Trjd/£ ) 



D - 



, J = 1,2, 



(2-83) 



For an enclosed harbor, of approximately rectangular planform with length 

 Jig, waves entering through a breakwater gap having a predominant period close 

 to one of those given by equation (2-83) for small values of j may cause 

 significant agitation unless some effective energy dissipation mechanism is 

 present. The addition of energy to the basin at the resonant (or excitation) 

 frequency (f^ = l/T^ is said to exoite the basin. 



Equation (2-83) was developed by assuming the end boundaries to be verti- 

 cal; however, it is still approximately valid so long as the end boundaries 

 remain highly reflective to wave motion. Sloping boundaries, such as beaches, 

 while usually effective energy dissipaters, may be significantly reflective if 

 the incident waves are extremely long. The effect of sloping boundaries and 

 their reflectivity to waves of differing characteristics is given in Section 

 V,4. 



Long-period resonant oscillations in large lakes and other large enclosed 

 bodies of water are termed seiches . The periods of seiches may range from a 

 few minutes up to several hours, depending on the geometry of the particular 

 basin. In general, these basins are shallow with respect to their length; 

 hence, tanh(Trjd/Jlg) in equation (2-83) becomes approximately equal to Trjd/£g 

 and 



1% 



T. = 



B 



1 



j (gd) 



1/2 



j = 1,2, 



(small values) 



(2-84) 



Equation (2-84) is termed Marian's equation. In natural basins, complex geom- 

 etry and variable depth will make the direct application of equation (2-84) 

 difficult; however, it may serve as a useful first approximation for enclosed 

 basins. For basins open at one end, different modes of oscillation exist 

 since resonance will occur when a node is at the open end of the basin and the 

 fundamental oscillation occurs when there is one-quarter of a wave in the 

 basin; hence, Jl^ = L/4 for the fundamental mode and T = 4£^//gd. In general 

 %\ = (2j - l)L/4, and 



^j (2j - 1) (gd) 



1/2 j = 1»2, 



(small values) 



(2-85) 



Note that higher modes occur when there are 3, 5, 

 of a wave within the basin. 



, 2j - 1, etc., quarters 



2-115 



